Local Reductions September 2014 Emanuele Viola Northeastern University
Papers: Local Reductions with Hamid Jahanjou and Eric Miles Succinct and explicit circuits for sorting and connectivity With Hamid Jahanjou and Eric Miles Short PCPs with projection queries With Eli Ben-Sasson
Theorem [Cook, Levin]: 3SAT is NP-complete
Theorem [Cook, Levin]: 3SAT is NP-complete M NTIME(t) reduction R : x R(x) = φ 3SAT M(x) = 1 R runs in time poly(t) (t = poly(n), t = 2 n etc.) Applications require to optimize (by themselves or both) - φ - Complexity of R
Optimizing φ (70s - 80s) [, Pippenger Fischer, Gurevich Shelah,...] φ = t log O(1) t Optimizing complexity of R. If reduction has resources polynomial in t, it is almost trivial Our focus: resources << t
Clause-explicit R R(i,x) = i-th clause of φ, e.g. ( y 15 V y 7 V y 8 ) i = log φ We will ignore x and focus on the map as a function of i, though dealing with x is not easy. Why care about explicitness?
Explicit R Succint-sat NEXP complete t = 2 n, φ = poly(t), R(i) run in time poly( i ) Lower bounds for SAT [Van Melkebeek, Fortnow, Lipton, Vigas] t = poly(n), φ = t log O(1) t, R(i) in time poly( i ), space O i Williams lower bounds from SAT/derandomization Lower bound against C (e.g., C = ACC 0 ), can use t = 2 n, φ = t log O(1) t, R(i) computable by C
Explicit R φ = poly(t), R AC 0 [Arora Steurer Wigderson] (or folklore) φ = t log O(1) t, R NC 1 [Ben-Sasson, Goldreich, Harsha, Sudan, Vadhan] (2005) Note: Williams ACC 0 lower bound uses workaround due to absence of more efficient reductions. More efficient reductions hard (perhaps impossible) Consequent drawbacks to be discussed shortly
Theorem [Jahanjou Miles V.] Reduce NTIME(t) to 3SAT via reduction R : φ = t log O(1) t Each output bit of R(i) depends on O(1) bits of i. (A.k.a. local, NC 0, junta). Note: R(i) = ( y 15 V y 7 V y 8 ) y 15 = log t = i bits; each bit depends on O(1) bits of i. Note: Local R cannot even compute i i+1
Outline Intro Consequences of local reductions Proof of local reductions PCP reductions
Warm-up consequence: SUCCINCT-3SAT, SUCCINCT-3COLOR, etc. remain NEXP complete even on instances represented by NC 0 circuits Slightly better ACC 0 lower bound
Consequence: Tighther connection between SAT algorithms and lower bounds NOTE: lower bound throughout means for f NEXP or E NP [W] gives lower bounds against size s, depth d from SAT algorithm for size s c, depth c d We only require SAT algorithm for size c s, depth d + c. This (and refinements) gives several new connections for classes of interest:
For each, new lower bound from SAT algorithm. Linear-size circuits Linear-size log-depth circuits [Valiant 1977] Linear-size series-parallel circuits [Valiant 1977] Quasi-polynomial SYM-AND circuits These can be related to assumptions about ksat
[W] Exponential-time hypothesis [Impagliazzo Paturi] false => linear-size circuits lower bound Our proof from previous result: Apply Cook-Levin. [JMV] Strong Exponential-time hypothesis false => linear-size series-parallel circuits lower bound [JMV] n c - SAT in time 2 n - ω n/log log n => linear-size log-depth circuits lower bound
Some tighther results [Ben-Sasson V., JMV] Unbounded-depth circuits: Lower bound for depth d <= SAT for depth d+1. Recall for general circuits a 3n lower bound is unknown. 3n lower bound from 3SAT in TIME(1.07) n non-boolean 3n lower bound from 3SAT in TIME(1.10) n Record: TIME(1.34) n
Do we simplify the proof [W] that NEXP is not in ACC 0? Recall that [W] uses as black-box previous reductions If instead use as black-box ours, the proof is more direct. In fact, for this application it suffices R Much easier to establish. AC 0 Independently, Kowalski and Van Melkebeek proved R AC 0
Outline Intro Consequences of local reductions Proof of local reductions PCP reductions
Background We reduce NTIME(t) to CIRCUIT-SAT C : (1) C = t log O(1) t (2) Given index i to gate, R(i) outputs type, and children with constant locality Pippenger Fischer oblivious simulation gives (1), but (2) hard Use alternative [Van Melkebeek], based on sorting networks (The idea of sorting is from Gurevich Shelah) Strangely little known!? Rediscovered by mini-poly-math class project at NEU
AND
That's why sorting matters! AND
Sorting network. This can be done quite efficiently, but O(1) locality unknown [Separate write-up, all that you need for AC 0 reduction] For constant locality, we instead use routing networks, as in PCP literature since Polischuck and Spielman With De Buijin graphs, computation very simple: children of i are i XOR CONSTANT (i rotated) XOR constant
Check circuits: Easy to obtain R running in linear space (= log C space). Theorem [JMV] For every C with linear-space R there is equivalent C', C' = poly C, with local R Technique [Ruzzo] New gates of C' are configurations of linear-space R. But Ruzzo does not aim or prove constant locality. Obtaining that is not trivial, as you can't check if a configuration is valid.
Problem: Given index to i-th configuration, need to compute index to (i+1) configuration Recall you cannot even compute i i+1
Problem: Given index to i-th configuration, need to compute index to (i+1) configuration Solution: Use routing networks in a different way. Instead of output of network being sorted order, it will be successor function. Config 2 Config 3 Config 1 Check Routing Config 1 Config 2 Config T
Outline Intro Consequences of local reductions Proof of local reductions PCP reductions
Note: This is for size n 3, much of what we saw earlier was for size O(n). > 10-year old problem: MAX-3SAT in time 2 n / n ω(1)? Equivalently, SAT of MAJ-AND 3 circuits Bottleneck for Williams' approach based on SAT algorithms. Needed for TC 0, threshold of threshold, etc.
Derandomization comes to rescue. MAJ-AND 3 and some other classes, can be derandomized. This suffices for lower bounds [W], using PCP reductions. Same considerations made earlier about Cook-Levin: 1) more efficient reduction => tighter connection 2) [W, Santhanam W] need workaround due to INefficiency of reductions.
[Ben-Sasson, Goldreich, Harsha, Sudan, Vadhan] Explicit PCP with t log O(1) t constraints, many queries [Mie] Improves queries to O(1). Theorem: [Ben-Sasson V.] Variant of [BGHSV] PCP: given index to constraint, variables (a.k.a. queries) are projections. Postprocess is a CNF [easy] Note: Projection queries were used in concurrent [W] lower bound for AC 0 SYM from #SAT. By above enough to derandomize (or SAT)
Consequence: Derandomizing (unbounded fan-in) depth d+2 circuits lower bound for depth d Example: depth-2 threshold lower bound still open.
Question: Improve number of queries to O(1), matching [Mie] How efficient PCP reductions? Constant locality?